On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern Richard Arratia Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 email: rarratia@math.usc.edu Submitted: July 27, 1999; Accepted: August 25, 1999. Abstract. Consider, for a permutation σ ∈S k ,thenumberF(n, σ) of permuta- tions in S n which avoid σ as a subpattern. The conjecture of Stanley and Wilf is that for every σ there is a constant c(σ) < ∞ such that for all n, F (n, σ) ≤ c(σ) n . All the recent work on this problem also mentions the “stronger conjecture” that for every σ, the limit of F (n, σ) 1/n exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity. We also discuss n-permutations, containing all σ ∈S k as subpatterns. We prove that this can be achieved with n = k 2 , we conjecture that asymptotically n ∼ (k/e) 2 is the best achievable, and we present Noga Alon’s conjecture that n ∼ (k/2) 2 is the threshold for random permutations. Mathematics Subject Classification: 05A05,05A16. 1. Introduction Consider, for a permutation σ ∈S k ,thesetA(n, σ) of permutations τ ∈S n which avoid σ as a subpattern, and its cardinality, F (n, σ):=|A(n, σ) |. Recall that “τ contains σ” as a subpattern means that there exist 1 ≤ x 1 <x 2 < ···<x k ≤ n such that for 1 ≤ i, j ≤ k, τ(x i ) <τ(x j ) if and only if σ(i) <σ(j).(1) An outstanding conjecture is that for every σ there is a finite constant c(σ)such that for all n, F (n, σ) ≤ c(σ) n . In the 1997 Ph.D. thesis of B´ona [2], supervised by The author thanks Noga Alon, B´ela Bollob´as, and Mikl´os B´ona for discussions of this problem. 1 2 the electronic journal of combinatorics 6 (1999), #N1 Stanley, this conjecture is attributed to “Wilf and Stanley [oral communication] from 1990.” All the recent work on this problem also mentions the “stronger conjecture” that for every σ, the limit of F (n, σ) 1/n exists and is finite. According to Wilf (private communication, 1999) the original conjecture was of this latter form. In this short note we give, as Theorem 1, a simple argument, involving subadditiv- ity, which shows that the two versions of the conjecture are equivalent. Here is some background information on the current status of the Stanley-Wilf conjecture. Represent σ ∈S k by the word σ(1) σ(2) ···σ(k). For the case of the increasing pattern, i.e the identity permutation, σ =12···k, the upper bound F (n, σ) ≤ ((k −1) 2 ) n is well known, and follows from the Robinson-Schensted-Knuth correspondence; also Regev [7] gives the asymptotics F (n, 12 ···k) ∼ λ k (k −1) 2n n k(k−2)/2 , with an explicit constant λ k . Simion and Schmidt [8] give a bijective proof that for each σ ∈S 3 , F(n, σ)= 1 n+1 2n n ; see also Knuth [6], section 2.2.1, exercises. For σ = 1342, B´ona [2] finds the explicit generating function for F(n, σ), showing that for all n, F (n, 1342) < 8 n , and lim F (n, 1342) 1/n = 8. Note in contrast that lim F(n, 1234) 1/n =9. B´ona observes that indeed, in all cases for which lim F(n, σ) 1/n is known explicitly, it is an integer! For the special class of “layered patterns,” such as σ = 67 345 12, B´ona [3] has shown that sup n F (n, σ) 1/n is finite. Alon and Friedgut [1] prove an upper bound for the general case which is tantalizingly close to the goal; they relate the problem to a result on generalized Davenport-Schinzel sequences from Klazar [5], and show that for every σ ∈S k there exists c(σ) < ∞ such that for all n, F (n, σ) ≤ c(σ) nγ ∗ (n) ,whereγ ∗ (n) is an extremely slowly growing function, given explicitly in terms of the inverse of the Ackermann function. Theorem 1. For every k ≥ 2 and σ ∈S k , for every m, n ≥ 1, F (m + n, σ) ≥ F (m, σ) F (n, σ)(2) and F (n, σ) ≥ 1; hence by Fekete’s lemma on subadditive sequences, c(σ) := lim n→∞ F (n, σ) 1/n ∈ [1, ∞] exists,(3) and ∀n ≥ 1,F(n, σ) ≤ c(σ) n . Proof. First we will show (2) by constructing, from an m-permutation and an n- permutation which avoid τ ,an(m + n)-permutation which avoids τ, injectively. Without loss of generality, we may assume that k precedes 1 in σ (since, with (·) r to denote the left-right reverse of a permutation, τ avoids σ iff τ r avoids σ r , and hence for all n, F(n, σ)=F (n, σ r ).) the electronic journal of combinatorics 6 (1999), #N1 3 Let τ ∈S m and τ ∈S n ,whereeachofτ and τ avoids σ.Letτ be the result of adding m to each symbol in the word for τ ,sothatτ isawordinwhicheachof the symbols m +1, ,m+ n appears exactly once. Consider the concatenation τ of τ with τ as a permutation, τ ∈S m+n . Clearly, τ avoids σ, establishing (2). [In detail, suppose to the contrary that τ contains σ, say at the k-tuple of positions given by 1 ≤ x 1 <x 2 < ··· <x k ≤ m + n. Recall that k precedes 1 in σ;say that σ(a)=1andσ(b)=k with 1 ≤ b<a≤ k, so that by (1), for 1 ≤ i ≤ k, τ(x a ) ≤ τ(x i ) ≤ τ(x b ). If x k ≤ m then τ contains σ,andifx 1 >mthen τ contains σ. If neither of these, then the x 1 ≤ m so that τ(x 1 ) ≤ m, hence τ(x a ) ≤ τ(x 1 ) ≤ m and therefore x a ≤ m; similarly x k >mso that τ(x k ) >m, hence τ(x b ) ≥ τ(x k ) >m and therefore x b >m, contradicting b<a.] Recalling that k precedes 1 in σ, the identity permutation in S n avoids σ and demonstrates that F (n, σ) ≥ 1 for every n ≥ 1. Fekete’s lemma [4], see also [9], is that if a 1 ,a 2 , ∈ R satisfy for all m, n ≥ 1, a m + a n ≤ a m+n , then lim n→∞ a n /n = inf n≥1 a n /n ∈ [−∞, ∞). Applying this with a n := −log F (n, σ) completes our proof. There exist [10] examples with σ, σ ∈S k ,withσ the identity permutation, and F (n, σ) >F(n, σ ), and B´ona [2], Theorem 4 shows that for all n ≥ 7, F(n, 1324) > F (n, 1234). Nevertheless, it is possible that for every k, the largest exponential growth rate is the (k −1) 2 achieved by the identity permutation. Conjecture 1. ($100.00) For all σ ∈S k and n ≥ 1, F (n, σ) ≤ (k −1) 2n . The problem of the shortest common superpattern. Define G(n, k) to be the number of permutations τ ∈S n which avoid at least one permutation in S k , i.e. G(n, k):=|∪ σ∈S k A(n, σ) |, where F (n, σ):=|A(n, σ) |. Simion and Schmidt [8], p. 398, give a formula for n! − G(n, 3), the number of n-permutations which contain all six patterns of length 3. In considering G(n, k), it is natural to consider the length m(k) of the shortest permutation which contains every σ ∈S k as a subpattern, i.e. to consider m(k):=min{n: G(n, k) <n! } =min{n: ∪ σ∈S k A(n, σ) = S n }. For a trivial lower bound on m(k), since τ ∈S n contains at most n k subpatterns, to contain every subpattern requires n k ≥ k!, hence lim inf k m(k)/k 2 ≥ 1/e 2 . Theorem 2. There exists an n-permutation, with n = k 2 , containing every k-permutation as a subpattern; i.e. m(k) ≤ k 2 . 4 the electronic journal of combinatorics 6 (1999), #N1 Proof. Consider the lexicographic order on [k] 2 as a one-to-one map specifying the ranks of the ordered pairs, i.e. let r :[k] 2 → [k 2 ], with (i, j) → (i − 1)k + j.Also consider the transposed lexicographic order t :[k] 2 → [k 2 ]givenbyt(i, j):=r(j, i). Consider the permutation τ ∈S k 2 given by τ = r ◦t −1 ; for example, with k =3,this is τ = 147258369. Then, clearly, τ contains every σ ∈S k as a subpattern. In detail, with the positions x 1 := t(σ(1), 1), , x k := t(σ(k),k)wehavex 1 < ···<x k and for m =1tok, τ(x m )=(r ◦ t −1 )(t(σ(m),m)) = r(σ(m),m)sothatτ(x a ) <τ(x b ) iff σ(a) <σ(b). Conjecture 2. As k →∞, m(k) ∼ (k/e) 2 . In contrast, from the known behavior of the length L n of the longest increasing subsequence, L n ∼ 2 √ n with high probability, one cannot hope to use random per- mutations to show that lim inf m(k)/k 2 ≤ (1/e) 2 . The probability that a random n-permutation does not contain every σ ∈S k as a subpattern is G(n, k)/n!. Define the threshold t(k)byt(k)=min{n: G(n, k)/n! ≤ 1/2}, so that trivially m(k) ≤ t(k), and hence lim inf t(k)/k 2 ≥ 1/4. Conjecture 3. (Noga Alon) The threshold length t(k), for a random permutation to contain all k-permutations with substantial probability, has t(k) ∼ (k/2) 2 . References [1] Alon, N., and Friedgut, E. (1999) On the number of permutations avoiding a given pattern. J. Combinatorial Theory, Ser. A, to appear [2] B´ona, M. (1997) Exact and asymptotic enumeration of permutations with subsequence condi- tions. Ph.D. Thesis, M.I.T. [3] B´ona, M. (1999) The solution of a conjecture of Stanley and Wilf for all layered patterns. J. Combinatorial Theory, Ser. A 85, 96-104. [4] Fekete, M. (1923) ¨ Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzz¨ahligen Koeffizienten. Math. Z. 17, 228-249. [5] Klazar, M. (1992) A general upper bound in extremal theory of sequences. Comment. Math. Univ. Carolin. 33, 737-746. [6] Knuth, D. E. (1968) The art of computer programming. Addison-Wesley, Reading, MA. [7] Regev, A. (1981) Asymptotic values for degrees associated with strips of Young diagrams. Adv. Math. 41, 115-136. [8] Simion, R., and Schmidt, F. W. (1985) Restricted permutations. European J. of Combinatorics 6, 383-406. [9] Steele, J. M. (1997) Probability theory and combinatorial optimization. CBMS-NSF regional conference series in applied mathematics 69. SIAM, Philidelphia, PA. [10] West, J. (1990) Permutations with forbidden subsequences; and stack sortable permutations. Ph.D. Thesis, MIT. . On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern Richard Arratia Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 email:. Friedgut, E. (1999) On the number of permutations avoiding a given pattern. J. Combinatorial Theory, Ser. A, to appear [2] B´ona, M. (1997) Exact and asymptotic enumeration of permutations with subsequence. permutation σ ∈S k ,thesetA(n, σ) of permutations τ ∈S n which avoid σ as a subpattern, and its cardinality, F (n, σ):= |A( n, σ) |. Recall that “τ contains σ” as a subpattern means that there exist 1